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coloring_nested_tire_graphs: add figure of the complete tire dual D*(T)

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Adds fig_complete_tire_dual.png next to Definition 1.13 of the
complete tire dual.  The figure uses the same m=6, k=4 spoke-only
tire as the partial-tire-dual figure (so the two figures can be
directly compared), with the n=6 outer leaves merged into a single
outer-face vertex v_out (blue hexagon, drawn outside the tire) of
degree 6, and the m=4 inner leaves merged into a single inner-face
vertex v_in (red hexagon, drawn at the centre of the inner cycle)
of degree 4.

Generator: experiments/draw_complete_tire_dual.py.

Paper grows from 8 to 9 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-25 19:28:19 -04:00
parent 93ae55bd42
commit 2b84513f83
7 changed files with 257 additions and 34 deletions
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papers/coloring_nested_tire_graphs/experiments/complete_tire_dual_example.png
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papers/coloring_nested_tire_graphs/experiments/draw_complete_tire_dual.py
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"""Draw the complete tire dual D*(T) on top of a small tire graph.
For a spoke-only tire with 2-connected O = C_k (no chords), D*(T) is
obtained from D(T) by merging:
- all B_out-leaves into a single outer-face vertex v_out (degree n);
- the B_in-leaves into a single inner-face vertex v_in (degree k),
because O has exactly one bounded interior face.
Equivalently, D*(T) is the planar dual of T.
"""
import math
import os
import sys
HERE = os.path.dirname(os.path.abspath(__file__))
sys.path.insert(0, HERE)
import matplotlib.pyplot as plt
import matplotlib.patches as patches
from tire_graph import random_tire, planar_positions
def draw_complete_tire_dual(tire, filename):
m, k = tire['m'], tire['k']
outer, inner = tire['outer'], tire['inner']
triangles = tire['triangles']
R_out, R_in = 1.0, 0.45
pos = planar_positions(tire, R_out=R_out, R_in=R_in)
fig, ax = plt.subplots(figsize=(9, 9))
# === Draw underlying tire graph faintly ===
outer_set = set(outer)
inner_set = set(inner)
for u, v in tire['edges']:
x1, y1 = pos[u]; x2, y2 = pos[v]
if u in outer_set and v in outer_set:
color = '#a8c9e8'; lw = 2.0
elif u in inner_set and v in inner_set:
color = '#e8a8a8'; lw = 2.0
else:
color = '#cccccc'; lw = 1.0
ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=1)
for v in outer:
x, y = pos[v]
ax.plot(x, y, 'o', color='#a8c9e8', markersize=14, zorder=2)
ax.annotate(str(v), (x, y), color='white', ha='center', va='center',
fontsize=8, fontweight='bold', zorder=3)
for v in inner:
x, y = pos[v]
ax.plot(x, y, 'o', color='#e8a8a8', markersize=12, zorder=2)
ax.annotate(str(v), (x, y), color='white', ha='center', va='center',
fontsize=7, fontweight='bold', zorder=3)
n_tri = len(triangles)
centroids = []
for tri in triangles:
v1, v2, v3 = tri
cx = (pos[v1][0] + pos[v2][0] + pos[v3][0]) / 3
cy = (pos[v1][1] + pos[v2][1] + pos[v3][1]) / 3
centroids.append((cx, cy))
# Identify which triangles are incident to a B_out edge (O-move type)
# vs incident to a B_in edge (I-move type).
from collections import defaultdict
bout_edges = {frozenset({outer[i], outer[(i+1) % m]}) for i in range(m)}
bin_edges = {frozenset({inner[j], inner[(j+1) % k]}) for j in range(k)}
bout_incident = [] # triangles incident to a B_out edge
bin_incident = [] # triangles incident to a B_in edge
for t_idx, tri in enumerate(triangles):
tri_edges = {frozenset({tri[0], tri[1]}),
frozenset({tri[1], tri[2]}),
frozenset({tri[0], tri[2]})}
if tri_edges & bout_edges:
bout_incident.append(t_idx)
if tri_edges & bin_edges:
bin_incident.append(t_idx)
print(f"O-move triangles (B_out incident): {bout_incident}")
print(f"I-move triangles (B_in incident): {bin_incident}")
# Place v_out outside the outer cycle (far to the upper right)
# and v_in at the origin (centroid of inner cycle).
v_out_pos = (1.55, 0.0) # well outside the outer cycle
v_in_pos = (0.0, 0.0) # center of inner cycle
# Identify dual cycle (interior dual edges)
edge_to_tris = defaultdict(list)
for t_idx, tri in enumerate(triangles):
e1 = frozenset({tri[0], tri[1]})
e2 = frozenset({tri[1], tri[2]})
e3 = frozenset({tri[0], tri[2]})
for e in (e1, e2, e3):
edge_to_tris[e].append(t_idx)
tri_adj = defaultdict(set)
for e, ts in edge_to_tris.items():
if len(ts) == 2:
t1, t2 = ts
tri_adj[t1].add(t2)
tri_adj[t2].add(t1)
# === Draw complete tire dual ===
DUAL_COLOR = '#7d3c98' # purple
OUT_COLOR = '#1f77b4' # blue (for v_out edges)
IN_COLOR = '#d62728' # red (for v_in edges)
# Interior dual edges (d_f to d_f)
for t1 in range(n_tri):
for t2 in tri_adj[t1]:
if t2 <= t1:
continue
x1, y1 = centroids[t1]
x2, y2 = centroids[t2]
ax.plot([x1, x2], [y1, y2], color=DUAL_COLOR, linewidth=2.0,
zorder=4)
# v_out edges (to all O-move d_f's)
for t_idx in bout_incident:
cx, cy = centroids[t_idx]
ax.plot([cx, v_out_pos[0]], [cy, v_out_pos[1]],
color=OUT_COLOR, linewidth=1.6, linestyle='--', zorder=4)
# v_in edges (to all I-move d_f's)
for t_idx in bin_incident:
cx, cy = centroids[t_idx]
ax.plot([cx, v_in_pos[0]], [cy, v_in_pos[1]],
color=IN_COLOR, linewidth=1.6, linestyle='--', zorder=4)
# Interior dual vertices d_f
for t_idx, (cx, cy) in enumerate(centroids):
ax.plot(cx, cy, 's', color=DUAL_COLOR, markersize=14, zorder=5)
ax.annotate(f"$d_{{{t_idx}}}$", (cx, cy), color='white',
ha='center', va='center', fontsize=7, fontweight='bold',
zorder=6)
# v_out (outer-face vertex)
ax.plot(v_out_pos[0], v_out_pos[1], 'h', color=OUT_COLOR,
markersize=20, zorder=5)
ax.annotate("$v_{\\mathrm{out}}$", v_out_pos, color='white',
ha='center', va='center', fontsize=9, fontweight='bold',
zorder=6)
# v_in (inner-face vertex)
ax.plot(v_in_pos[0], v_in_pos[1], 'h', color=IN_COLOR,
markersize=18, zorder=5)
ax.annotate("$v_{\\mathrm{in}}$", v_in_pos, color='white',
ha='center', va='center', fontsize=8, fontweight='bold',
zorder=6)
# Legend
legend_items = [
plt.Line2D([], [], marker='o', color='w', markerfacecolor='#a8c9e8',
markersize=12, label='$B_{\\mathrm{out}}$ vertex (tire)'),
plt.Line2D([], [], marker='o', color='w', markerfacecolor='#e8a8a8',
markersize=11, label='$B_{\\mathrm{in}}$ vertex (tire)'),
plt.Line2D([], [], marker='s', color='w', markerfacecolor=DUAL_COLOR,
markersize=12, label='$d_f$ (annular face dual vertex)'),
plt.Line2D([], [], marker='h', color='w', markerfacecolor=OUT_COLOR,
markersize=14,
label='$v_{\\mathrm{out}}$ (outer-face vertex, deg ${%d}$)' % m),
plt.Line2D([], [], marker='h', color='w', markerfacecolor=IN_COLOR,
markersize=13,
label='$v_{\\mathrm{in}}$ (inner-face vertex, deg ${%d}$)' % k),
plt.Line2D([], [], color=DUAL_COLOR, linewidth=2.0,
label='interior dual edge ($d_f$–$d_{f^\\prime}$)'),
plt.Line2D([], [], color=OUT_COLOR, linewidth=1.6, linestyle='--',
label='dual edge to $v_{\\mathrm{out}}$'),
plt.Line2D([], [], color=IN_COLOR, linewidth=1.6, linestyle='--',
label='dual edge to $v_{\\mathrm{in}}$'),
]
ax.legend(handles=legend_items, loc='upper left',
bbox_to_anchor=(1.02, 1.0), fontsize=9, frameon=False)
ax.set_xlim(-1.35, 2.05)
ax.set_ylim(-1.35, 1.35)
ax.set_aspect('equal')
ax.axis('off')
ax.set_title(
f'Complete tire dual $D^{{*}}(T)$ (m={m}, k={k}, spoke-only)\n'
f'underlying tire faint; $|V| = {n_tri + 2}$, $|E| = {2*n_tri}$',
fontsize=12)
plt.savefig(filename, dpi=160, bbox_inches='tight')
plt.close()
print(f"wrote {filename}")
def main():
tire = random_tire(m=6, k=4, n_chords=0, seed=3)
out = os.path.join(HERE, 'complete_tire_dual_example.png')
draw_complete_tire_dual(tire, out)
if __name__ == '__main__':
main()
papers/coloring_nested_tire_graphs/fig_complete_tire_dual.png
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papers/coloring_nested_tire_graphs/paper.aux
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\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}}
\newlabel{def:complete-tire-dual}{{1.13}{7}}
\citation{Tait1880}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The complete tire dual $D^{\ast }(T)$ (purple squares + face hexagons) drawn on top of the same $m = 6$, $k = 4$ spoke-only tire graph as Figure\nonbreakingspace 3\hbox {}. The ten annular-face vertices $d_f$ form a $10$-cycle (solid purple); the $n$ outer leaves of $D(T)$ have been merged into a single outer-face vertex $v_{\mathrm {out}}$ (blue hexagon, drawn outside the tire) of degree $n = 6$, and the $m$ inner leaves into a single inner-face vertex $v_{\mathrm {in}}$ (red hexagon, at the centre of the inner cycle) of degree $m = 4$. Total $|V(D^{\ast }(T))| = 12$ and $|E(D^{\ast }(T))| = 20$.}}{8}{}\protected@file@percent }
\newlabel{fig:complete-tire-dual-example}{{4}{8}}
\newlabel{prop:tait-tire-complete}{{1.14}{8}}
\newlabel{rem:tait-construction}{{1.15}{8}}
\newlabel{rem:tait-octahedron}{{1.16}{8}}
\bibcite{Tait1880}{1}
\bibcite{bauerfeld-pds}{2}
\newlabel{tocindent-1}{0pt}
@@ -26,8 +31,5 @@
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\newlabel{prop:tait-tire-complete}{{1.14}{8}}
\newlabel{rem:tait-construction}{{1.15}{8}}
\newlabel{rem:tait-octahedron}{{1.16}{8}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent }
\gdef \@abspage@last{8}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent }
\gdef \@abspage@last{9}
papers/coloring_nested_tire_graphs/paper.log
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@@ -213,37 +213,43 @@ File: fig_partial_tire_dual.png Graphic file (type png)
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papers/coloring_nested_tire_graphs/paper.tex
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@@ -508,6 +508,22 @@ vertex per face of $T$ (annular triangle, outer face, or bounded
interior face of $O$), one dual edge per edge of $T$.
\end{definition}
\begin{figure}[h]
\centering
\includegraphics[width=0.82\textwidth]{fig_complete_tire_dual.png}
\caption{The complete tire dual $D^{\ast}(T)$ (purple squares + face
hexagons) drawn on top of the same $m = 6$, $k = 4$ spoke-only tire
graph as Figure~\ref{fig:partial-tire-dual-example}. The ten
annular-face vertices $d_f$ form a $10$-cycle (solid purple); the
$n$ outer leaves of $D(T)$ have been merged into a single outer-face
vertex $v_{\mathrm{out}}$ (blue hexagon, drawn outside the tire) of
degree $n = 6$, and the $m$ inner leaves into a single inner-face
vertex $v_{\mathrm{in}}$ (red hexagon, at the centre of the inner
cycle) of degree $m = 4$. Total $|V(D^{\ast}(T))| = 12$ and
$|E(D^{\ast}(T))| = 20$.}
\label{fig:complete-tire-dual-example}
\end{figure}
\begin{proposition}[Tait correspondence on the complete tire dual]
\label{prop:tait-tire-complete}
Let $T$ be a tire graph. Then the number of non-equivalent proper